If ` a^(3) + 3 a^(2) + 9 a = 1 ` then what is the value of ` a^(3) + (3)/( a)` ?

Options

A. 31
B. 26
C. 28
D. 24

Answer

To solve the equation \( a^3 + 3a^2 + 9a = 1 \) and find the value of \( a^3 + \frac{3}{a} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ a^3 + 3a^2 + 9a = 1 \]

### Step 2: Divide the entire equation by \( a \)
To manipulate the equation, we can divide each term by \( a \) (assuming \( a \neq 0 \)):
\[ \frac{a^3}{a} + \frac{3a^2}{a} + \frac{9a}{a} = \frac{1}{a} \]
This simplifies to:
\[ a^2 + 3a + 9 = \frac{1}{a} \]

### Step 3: Multiply the equation by 3
Next, we multiply the entire equation by 3:
\[ 3(a^2 + 3a + 9) = 3 \cdot \frac{1}{a} \]
This gives us:
\[ 3a^2 + 9a + 27 = \frac{3}{a} \]

### Step 4: Add \( a^3 \) to both sides
Now, we add \( a^3 \) to both sides of the equation:
\[ a^3 + 3a^2 + 9a + 27 = \frac{3}{a} + a^3 \]
Since we know from the original equation that \( a^3 + 3a^2 + 9a = 1 \), we can substitute:
\[ 1 + 27 = \frac{3}{a} + a^3 \]

### Step 5: Simplify the left side
Now we simplify the left side:
\[ 28 = \frac{3}{a} + a^3 \]

### Step 6: Isolate \( a^3 + \frac{3}{a} \)
Thus, we can express \( a^3 + \frac{3}{a} \) as:
\[ a^3 + \frac{3}{a} = 28 \]

### Conclusion
Therefore, the value of \( a^3 + \frac{3}{a} \) is:
\[ \boxed{28} \]